On the variation of the fractional mean curvature under the effect of C^{1, \alpha} perturbations

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December 2015, 35(12): 5769-5786. doi: 10.3934/dcds.2015.35.5769

On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

1.	Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Received March 2014 Published May 2015

Abstract
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In this brief note we study how the fractional mean curvature of order $s \in (0, 1)$ varies with respect to $C^{1, \alpha}$ diffeomorphisms. We prove that, if $\alpha > s$, then the variation under a $C^{1, \alpha}$ diffeomorphism $\Psi$ of the $s$-mean curvature of a set $E$ is controlled by the $C^{0, \alpha}$ norm of the Jacobian of $\Psi$. When $\alpha = 1$ we discuss the stability of these estimates as $s \rightarrow 1^-$ and comment on the consistency of our result with the classical framework.

Keywords: $C^{1, Fractional mean curvature, \alpha}$ diffeomorphisms, fractional Laplacian, quantitative Implicit Function Theorem., non-local equations, $s$-perimeter.

Mathematics Subject Classification: Primary: 35R11, 28A75, 49Q05; Secondary: 26B1.

Citation: Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769

References:

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[14]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Adv. Math., 248* (2013), 843. doi: 10.1016/j.aim.2013.08.007. Google Scholar*
[15]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136* (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar*
[16]	S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0,, Discrete Contin. Dyn. Syst., 33 (2013), 2777. doi: 10.3934/dcds.2013.33.2777. Google Scholar
[17]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces,, \emph{J. Reine Angew. Math.}, (2015). doi: 10.1515/crelle-2015-0006. Google Scholar
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[19]	M. Ludwig, Anisotropic fractional perimeters,, J. Differential Geom., 96 (2014), 77. Google Scholar
[20]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Ration. Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230. Google Scholar
[21]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285. Google Scholar
[22]	G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm,, Ann. Mat. Pura Appl., 192* (2013), 673. doi: 10.1007/s10231-011-0243-9. Google Scholar*
[23]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479. doi: 10.1016/j.anihpc.2012.01.006. Google Scholar
[24]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2,, Calc. Var. Partial Differential Equations, 48 (2013), 33. doi: 10.1007/s00526-012-0539-7. Google Scholar
[25]	E. Valdinoci, A fractional framework for perimeters and phase transitions,, Milan J. Math., 81 (2013), 1. doi: 10.1007/s00032-013-0199-x. Google Scholar

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References:

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals,, Manuscripta Math., 134* (2011), 377. doi: 10.1007/s00229-010-0399-4. Google Scholar*
[2]	N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature,, Numer. Funct. Anal. Optim., 35 (2014), 793. doi: 10.1080/01630563.2014.901837. Google Scholar
[3]	G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators,, J. Math. Anal. Appl., 428* (2015), 1225. doi: 10.1016/j.jmaa.2015.04.002. Google Scholar*
[4]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem,, Math. Ann., 310* (1998), 527. doi: 10.1007/s002080050159. Google Scholar*
[5]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies,, European J. Appl. Math., 9 (1998), 261. doi: 10.1017/S0956792598003453. Google Scholar
[6]	B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609. Google Scholar
[7]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions,, Trans. Amer. Math. Soc., 367* (2015), 911. doi: 10.1090/S0002-9947-2014-05906-0. Google Scholar*
[9]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces,, Comm. Pure Appl. Math., 63 (2010), 1111. doi: 10.1002/cpa.20331. Google Scholar
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar
[11]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. doi: 10.1002/cpa.20274. Google Scholar
[12]	L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200* (2011), 59. doi: 10.1007/s00205-010-0336-4. Google Scholar*
[13]	L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces,, Calc. Var. Partial Differential Equations, 41 (2011), 203. doi: 10.1007/s00526-010-0359-6. Google Scholar
[14]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Adv. Math., 248* (2013), 843. doi: 10.1016/j.aim.2013.08.007. Google Scholar*
[15]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136* (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar*
[16]	S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0,, Discrete Contin. Dyn. Syst., 33 (2013), 2777. doi: 10.3934/dcds.2013.33.2777. Google Scholar
[17]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces,, \emph{J. Reine Angew. Math.}, (2015). doi: 10.1515/crelle-2015-0006. Google Scholar
[18]	N. S. Landkof, Foundations of Modern Potential Theory,, Springer-Verlag, (1972). Google Scholar
[19]	M. Ludwig, Anisotropic fractional perimeters,, J. Differential Geom., 96 (2014), 77. Google Scholar
[20]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Ration. Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230. Google Scholar
[21]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285. Google Scholar
[22]	G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm,, Ann. Mat. Pura Appl., 192* (2013), 673. doi: 10.1007/s10231-011-0243-9. Google Scholar*
[23]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479. doi: 10.1016/j.anihpc.2012.01.006. Google Scholar
[24]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2,, Calc. Var. Partial Differential Equations, 48 (2013), 33. doi: 10.1007/s00526-012-0539-7. Google Scholar
[25]	E. Valdinoci, A fractional framework for perimeters and phase transitions,, Milan J. Math., 81 (2013), 1. doi: 10.1007/s00032-013-0199-x. Google Scholar

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